Question

Prove that twice the area if the parallelogram id equal to the area of another parallelogram formed by taking as it's adjacent side the diagnosis of the former parallelogram

Answer 1

Step 1

Let ABCDABCD be a parallelogram .

Consider the parallelogram formed by its diagonals AC−→−andBD−→−AC→andBD→ as adjacent sides

Step 2

The vector are of such a parallelogram would be AC−→−×BD−→−=(AB−→−+BC−→−)×(BA−→−+AD−→−)AC→×BD→=(AB→+BC→)×(BA→+AD→)

=AB−→−×BA−→−+AB−→−×AD−→−+BC−→−×BA−→−+BC−→−×AD−→−=AB→×BA→+AB→×AD→+BC→×BA→+BC→×AD→

=0→+AB−→−×AD−→−−BC−→−×AB−→−+AD−→−×AD−→−=0→+AB→×AD→−BC→×AB→+AD→×AD→

=AB−→−×AD−→−−AD−→−×AB−→−=AB→×AD→−AD→×AB→

=AB−→−×AD−→−+AB−→−×AD−→−=AB→×AD→+AB→×AD→

=2AB−→−×AD−→−=2=2AB→×AD→=2 ( vector area of parallelogram ABCDABCD )

Hence proved